CONTOUR INTEGRALS
AND THEIR APPLICATIONS
Wayne Lawton
Department of Mathematics
National University of Singapore
S14-04-04, matwml@nus.edu.sg
http://math.nus.edu.sg/~matwml
ARYABHATA
characterized the set { (x, y) } of integer solutions of the equation
ax  by  1
where a and b are integers. Clearly this equation admits a
solution if and only if a and b have no common factors other
than 1, -1 (are relatively prime) and then Euclid’s algorithm
gives a solution. Furthermore, if (x,y) is a solution then the set
of solutions is the infinite set
{ ( x  kb, y  ka) : k is an integer }
Van der Warden, Geometry and Algebra in Ancient Civilizations,
Springer-Verlag, New York, 1984.
BEZOUT
investigated the polynomial version of this equation
P1Q1  P2Q2  1
P
P
Clearly this equation has a solution iff
and
have no
1
2
common roots and then Euclid’s algorithm gives a solution.
Bezout identities in general rings arise in numerous areas of
mathematics and its application to science and engineering:
Algebraic Polynomials: control, Quillen-Suslin Theorem
Laurent Polynomials: wavelet, splines, Swan’s Theorem
H_infinity: the Corona Theorem
Entire Functions: distributional solutions of systems of PDE’s
Matrix Rings: control, signal processing
E. Bezout, Theorie Generale des Equations Algebriques,
Paris, 1769.
INEQUALITY CONSTRAINTS
P1 , P2 are  0 on the unit circle T
then  LP Q1 , Q2  0 on T and P1Q1  P2Q2  1
Proof. Let LP B1 , B2 real on T with P1B1  P2 B2  1
1 that is real on T with
Choose a LP G  (P1  P2 )
Theorem If RPLP
1
[G - (P1  P2 ) ][1  Bi (P1  P2 )]  (P1  P2 )
1
Q
G

B
[
1

G
(
P

P
)]




1
1
1
2
then choose
Q   G  B [1  G(P  P )]
2
1
2 
 2 
W.Lawton & C.Micchelli, Construction of conjugate
quadrature filters with specified zeros, Numerical
Algorithms, 14:4 (1997) 383-399
W.Lawton & C.Micchelli, Bezout identities with inequality
constraints, Vietnam J. Math. 28:2(2000) 97-126
UPPER LENGTH BOUNDS
Theorem
  min{|    |: P1 ( )  P2 ( )  0}
L  min{max{(P1 ),  ( P2 )}} B  min{max{(Q1 ), (Q 2 )}}
There exists G : (0, )  Z   Z  with B  G (  , L)
Furthermore, for fixed

G : (  , L)  O ( L 5
5
L / 2 256 L2 / 2
(

)
), L  
and for fixed L
G : (  , L)  O( 
Proof: Uses resultants.
-L2 / 4
log  ),   0
1
LOWER LENGTH BOUNDS
Theorem For any positive integer n, there exist LP
P1 , P2
with
(P1 )  (P2 )  4n
and
B     4n
4
1
Proof: See VJM paper.
Question: Are there better ways to obtain bounds that
‘bridge the gap’ between the upper and lower bounds
CONTOUR INTEGRAL
representation for the Bezout identity is given by
Theorem Let  , are a disjoint contours and the
1
2
interior  of  contains all roots of P3-k, and
k
k
Pk , k  1,2 then for
z  C \  where   1  2
excludes all roots of
1
Bk ( z )  P3k ( z )  [2 i ( z   )T( )] d
k
are LP, real on T, and satisfy the Bezout identity.
Proof Follows from the residue calculus.
SOLUTION BOUNDS
~
M(, P1 , P2 ) |  |
Lemma  
where
~
2  d(, ) m(, T)
  max {| Bk ( z) | : z  T, k  1,2}
~ a contour that is disjoint from
  {0}

and whose (annular) interior contains T
~
~
M(, P1 , P2 )  max {| Pk ( z ) |: z  , k  1,2}
~
~
d(, )  min {|    |:   ,   }
m(, T)  min {| T( ) |:  }
|  |  length ()
CONTOUR CONSTRUCTION
1

(
z
)


P
(

(
z
))

P
(
z
)
Since P  0 on T,
z
k
hence if  are  -invariant contours then it suffices
k
to consider these quantities inside of the unit disk D.
For k=1,2 let D  union of open disks of radius
k
 1
4 8L
  min{ , } centered at zeros of P3-k in D
and D be the disk of this radius centered at 0.
0
E k  Dk if Dk  D0   else E k  Dk  D0
Fk  Ek  D, ˆ k  Fk   D, k  ˆ k   (ˆ k )
Theorem
  ( L  1)( ) 
9 L/2
2
2 L  2
CONCLUSIONS AND EXTENSIONS
The contour integral method provide sharper bounds for

and therefore for B than the resultant method but
sharper bounds are required to ‘bridge the gap’.
Contour integrals for BI with n > terms are given by
1
T ( z)
Bk ( z ) 
[
2

i
(
z


)
T
(

)]
d


(n - 1)Pk ( z ) k
where  encloses all zeros of T except for those of P
k
k
Residue current integrals give multivariate versions